A three-dimensional nonlinear viscous mathematical model with free moving boundaries will be introduced and a novel numerical method will be developed to solve the model to study blood flow in collapsible carotid arteries with stenoses. The numerical method uses solutions of the longwave asymptotic expansions as the numerical initial condition and a boundary-iterative method to find the unknown moving boundary and the flow velocity and pressure. The laboratory experiments and in vivo measurements from concurrent project by Dr. Ku at Georgia Tech will provide data for the formulation and verification of the mathematical model. The validated model together with its numerical solutions will form a basis for many further investigations of the collapsing process. While physiological dimensions of the carotid arteries and related parameter ranges will be used in the computations and experiments, the methods developed in this project will be equally applicable to a broad range of applications where free moving boundaries are involved.